Tuesday, February 3, 2015

Mach Number Versus Speed and Velocity

I've just a quick post today clearing up a simple misunderstanding I see all the time in amateur/layperson discussions of AEROnautics (taking a break from spaceflight for just a little bit). This is the difference between the mach number and speed/velocity of an aircraft. Quite frequently people confuse mach number for speed--e.g. talking about how fast or agile an aircraft is by talking about what mach number it can fly at--when the truth is a bit more complicated than this.

While the meaning of an aircraft (or rocket's) speed is clear to most people, its mach number is often treated as an equivalent substitute. Saying you're moving supersonically is a euphemism for going fast. But the mach number of an aircraft--and which side of the sonic boundary it's on-- is equal not to just the the speed of the aircraft but rather the speed of the aircraft divided by the local speed of sound. As an equation this is written

M = V/a
where M is the mach number, V is the speed, and a is the speed of sound. At first glance, especially to those new to aerospace, this may seem pointless; the speed of sound is constant, right? Well, actually no, it isn't. The speed of sound in a fluid varies as a function of three parameters: (1) the temperature of a fluid, (2) something called the specific gas constant of a fluid and (3) something called the specific heat ratio of the fluid. The first parameter has to do with the actual state of the fluid, while the second two only have to do with what kind of fluid is used and therefore are true constants for the atmosphere; at least to a good approximation.

Most people know that as you go further upward into the atmosphere, the pressure decreases. This is because there's less air molecules the further you go from the surface of the earth and thus there's less of a force per unit area pushing on you. What many people forget is that temperature too changes, as can be seen in the image below.

In everyday, low altitude flying, this change in temperature is not enough to significantly change the speed of sound, and thus in principle the mach number might be okay to use for a rough speed comparison, but there are actually only very few low altitude planes where the mach number matters, mostly because the speed of sound is so much faster than the aircraft at sealevel. At higher altitudes the speed of sound drops off. At still higher altitudes it grows again, drops off again, and finally grows strongly as you exit the planet. Thus the altitude an aircraft is flying at strongly influences the meaning of its mach number with respect to its velocity! 

A Boeing 747, for instance, cruises at roughly 20-30,000 ft--safely within relatively sealevel like conditions. Its mach number will be a good approximate indication of how fast it would be going at sealevel, but an SR-71 cruises at 85,000 ft, where the local speed of sound is almost 10% less! This means the mach number of the SR-71 would be greater than its mach number at the same speed at sealevel. There may even be flight regimes where the aircraft is operating supersonically in the altitude it was designed for and subsonically in a lower altitude. 

Other than speed, what is the mach number good for? Well, all sorts of things. In compressible fluids, like air, the equations governing aerodynamic and thermodynamic behavior are strongly related to the mach number, and the way they affect the structure and forces on an aircraft change considerably as it breaks the sound barrier. For instance, the heating which takes place behind a supersonic shockwave is significant, and thus an aircraft flying supersonically must have a well designed nosecone if it is to withstand this heating (since the nose is for all intents and purposes behind a supersonic shockwave as the plane is in flight). 

So next time someone quotes a mach number at you when talking about how fast an aircraft is going, ask them if it is a high altitude or low altitude aircraft. A large mach number for a high altitude aircraft may not correspond to a hugely greater speed than a low (or even subsonic) mach number for a lower altitude aircraft. 

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